Mass action models

Mass-action model of surfactant micelle formation was used for development of the conceptual retention model in micellar liquid chromatography. The retention model is based upon the analysis of changing of the sorbat microenvironment in going from mobile phase (micellar surfactant solution, containing organic solvent-modifier) to stationary phase (the surfactant covered surface of the alkyl bonded silica gel) according to equation ... 

Fig. 4 Reaction of benzoic anhydride in CTAOH , no added NaOH O, 0.01 M added NaOH , 0.02 M added NaOH. The lines are calculated using the mass-action model. (Reprinted with permission of the American Chemical Society)...

Despite these uncertainties values of kM for reactions of hydroxide ion in CTAOH and mixtures of CTABr or CTAC1 with NaOH calculated using the ion-exchange or mass-action models agree reasonably well, and some examples are given in Table 3. 

The symbols, IE or M A indicate that counterion binding was calculated using the ion exchange or mass action models and ST that the micelle was assumed to be saturated with counterion. 

Interactions between cationic micelles and uni- and divalent anions have been treated quantitatively by solving the Poisson-Boltzmann equation in spherical symmetry and considering both Coulombic and specific attractive forces. Predicted rate-surfactant profiles are similar to those based on the ion-exchange and mass-action models (Section 3), but fit the data better for reactions in solutions containing divalent anions (Bunton, C. A. and Moffatt, J. R. (1985) J. Phys. Chem. 1985, 89, 4166 1986,90, 538). 

To simulate the overall network behavior, the power-law formalism is applied in two different ways. Within a generalized mass-action model (GMA), each biochemical interconversion is modeled with a power-law term, resulting in a differential equation analogous to Eq. (5)... 

In contrast to generalized mass-action models, an S-system model is obtained by lumping (or aggregating) all synthesizing and consuming reactions of each metabolite into a single power-law term, respectively. The mathematical structure of a S-System is independent of the complexity of the network. For any metabolite. S, -, we obtain... 

Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state v° and S°. To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. 

The mass action model (MAM) for binary ionic or nonionic surfactants and the pseudo-phase separation model (PSM) which were developed earlier (I EC Fundamentals 1983, 22, 230 J. Phys. Chem. 1984, 88, 1642) have been extended. The new models include a micelle aggregation number and counterion binding parameter which depend on the mixed micelle composition. Thus, the models can describe mixtures of ionic/nonionic surfactants more realistically. These models generally predict no azeotropic micellization. For the PSM, calculated mixed erne s and especially monomer concentrations can differ significantly from those of the previous models. The results are used to estimate the Redlich-Kister parameters of monomer mixing in the mixed micelles from data on mixed erne s of Lange and Beck (1973), Funasaki and Hada (1979), and others. 

These four equations reduce to the equations of Ref. (11) if N and are fixed. The micelle concentration is then calculated from a mass action model expression as before, where (17)... 

A mass action model (MAM) with monodisperse aggregation number N which depends on the micelle mole fraction x and the counterion binding parameter /3(x) has been developed for binary surfactants either ionic/ionic or nonionic/ionic. 

In the pure liquid state or In the micellar form ( ). The parameters, derived from the mass-action model using data from the literature (16,23), are summarized In Table II. The curves A shown In... 

Table II. Parameters from the Mass-Action Model for the Binary System at 25°C...

To this point, only models based on the pseudo—phase separation model have been discussed. Mixed micelle models utilizing the mass action model may be necessary for micelles with small aggregation numbers, as demonstrated by Kamrath and Franses ( ). However, even for large micelles, the fundamental basis for the pseudophase separation model needs to be examined. In micelles, how much solvent or how many counterions (bound or in the electrical double layer) should be included in the micellar pseudo-phase is unclear. The difficulty is normally surmounted by assuming that the pseudo—phase consists of only the surfactant components i.e., solvent or counterions are ignored. The validity of treating the micelle on a surfactant—oniy basis has not been verified. Funasaki and Hada (22) have questioned the thermodynamic consistency of such an approach. 

The thermodynamics of micelle formation has been studied extensively. There is for example a mass action model (Wennestrdm and Lindman, 1979) that assumes that micelles can be described by an aggregate Mm with a single aggregation number m, so that the only descriptive equation is mMi Mm. A more complex form assumes the multiple equilibrium model, allowing aggregates of different sizes to be in equilibrium with each other (Tanford, 1978 Wennestrdm and Lindman, 1979 Israelachvili, 1992). 

In the mass action model the micellar system can be described by only one parameter, and despite this simplicity, a good qualitative description of the main physical properties is obtained, for example the onset of cmc (critical micelle concentration), as shown in Figure 9.7. Notice that the formation of micelles becomes appreciable only at the cmc, and after that, by increasing further the surfactant concentration, all added surfactant is transformed directly into micelles, so that the surfactant concentration in solution remains constant at the level of cmc. 

Following this, the thermodynamic arguments needed for determining CMC are discussed (Section 8.5). Here, we describe two approaches, namely, the mass action model (based on treating micellization as a chemical reaction ) and the phase equilibrium model (which treats micellization as a phase separation phenomenon). The entropy change due to micellization and the concept of hydrophobic effect are also described, along with the definition of thermodynamic standard states. 

We see in Section 8.8 that surfactants undergo aggregation in nonaqueous solvents also, but the degree of aggregation is very much less (n < 10), and the threshold for aggregation is far less sharp than in water. The mass action model for micellization seems preferable for nonaqueous systems. 

Two principal approaches have been used to describe the thermodynamics of surfactant solutions — the pseudo-phase model and the mass action model. 

The Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, f3. The reaction isdd... 

The lines through the data points in Figures 18.11 and 18.13 are fits of the mass action model to the experimental results. The agreement is excellent when we consider that only three variable parameters are required to fit each of the thermodynamic properties.hh... 

Several models have been developed to interpret micellar behavior (Mukerjee, 1967 Lieberman et al., 1996). Two models, the mass-action and phase-separation models are described here in mor detail. In the mass-action model, micelles are in equilibrium with the unassociated surfactant or monomer. For nonionic surfactants with an aggregation numb itbfe mass-action model predicts thatn molecules of monomeric nonionized surfactaStajeact to form a micelleM ... 

Contrary to the preceding treatment the so-called mass-action model develops apparently more naturally from the application of the mass action law applied to the overall aggregation process... 

Generalizing, one could state that the mass-action model simulates a cooperative (all or nothing) process with respect to large n values. This model is somewhat... 

Summarizing the statements of these three most commonly used models, it appears that the so-called mass action and phase-separation models simulate a third condition which must be fulfilled with respect to the formation of micelles a size limiting process. The latter is independent of the cooperativity and has to be interpreted by a molecular model. The limitation of the aggregate size in the mass action model is determined by the aggregation number. This is, essentially, the reason that this model has been preferred in the description of micelle forming systems. The multiple equilibrium model as comprised by the Eqs. (10—13) contains no such size limiting features. An improvement in this respect requires a functional relationship between the equilibrium constants and the association number n, i.e.,... 

From this equation, we can construct an example that is extremely useful in visually examining receptor data, indeed any data fitting the simple mass action model of Eq. (19.1). This means that a toxicant under study is competing with the radioligand for one, and only one, population of sites. By definition, the Hill slope (often called nH) must equal 1. It turns out that by memorizing the numbers 9 and 91, one can do a very useful preliminary analysis of data fit to such a mass-action model. The reason for this is as follows ... 

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